Parametric gates depend on a continuous parameter $\theta$, enabling tunable operations. Essential for variational algorithms and pulse control.

$$R_X(\theta) = e^{-i\theta X/2} = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

$$R_Y(\theta) = e^{-i\theta Y/2} = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

$$R_Z(\theta) = e^{-i\theta Z/2} = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$

Rotation by $\theta$ around specified axis. Key property: rotating by $2\pi$ returns to original state (periodicity).

The general U gate has three parameters:

$$U(\theta, \phi, \lambda) = \begin{pmatrix} \cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\ e^{i\phi}\sin(\theta/2) & e^{i(\phi+\lambda)}\cos(\theta/2) \end{pmatrix}$$

Can represent any single-qubit unitary. Implemented in most quantum systems.

$$XX(\theta) = e^{-i\theta X_1 X_2/2}$$

$$YY(\theta) = e^{-i\theta Y_1 Y_2/2}$$

$$ZZ(\theta) = e^{-i\theta Z_1 Z_2/2}$$

Controlled rotation between two qubits. Common in superconducting qubits via flux tuning.

Parametric gates enable training: optimize parameters $\theta_i$ to minimize a cost function:

$$\min_{\{\theta_i\}} \langle \psi(\{\theta_i\}) | H | \psi(\{\theta_i\}) \rangle$$

Gradient-based optimization uses parameter shift rule or automatic differentiation.